## 金融代写|波动率模型代写Market Volatility Modelling代考|COVID19

statistics-lab™ 为您的留学生涯保驾护航 在代写波动率模型Volatility Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写波动率模型Volatility Modelling代写方面经验极为丰富，各种代写波动率模型Volatility Modelling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Infinitesimal Generators and Associated Martingales

For simplicity we first consider a time-homogeneous diffusion process $\left(X_t\right)$ that solves the stochastic differential equation
$$d X_t=\mu\left(X_t\right) d t+\sigma\left(X_t\right) d W_t .$$
Let $g$ be a twice continuously differentiable function of the variable $x$ with bounded derivatives, and define the differential operator $\mathcal{L}$ acting on $g$ according to
$$\mathcal{L}{\mathcal{G}}(x)-\frac{1}{2} \sigma^2(x) g^{\prime \prime}(x)+\mu(x) g^{\prime}(x) .$$ In terms of $\mathcal{L}$, Itô’s formula (1.16) gives $$d g\left(X_t\right)=\mathcal{L} g\left(X_t\right) d t+g^{\prime}\left(X_t\right) \sigma\left(X_t\right) d W_t,$$ which shows that $$M_t=g\left(X_t\right)-\int_0^t \mathcal{L} g\left(X_s\right) d s$$ defines a martingale. Consequently, if $X_0=x$, we obtain $$\mathbb{E}\left{g\left(X_t\right)\right}=g(x)+\mathbb{E}\left{\int_0^t \mathcal{L} g\left(X_s\right) d s\right} .$$ Under the assumptions made on the coefficients $\mu$ and $\sigma$ and on the function $g$, the Lebesgue dominated convergence theorem is applicable and gives \begin{aligned} \left.\frac{d}{d t} \mathbb{E}\left{g\left(X_t\right)\right}\right|{t=0} & =\lim {t \downarrow 0} \frac{\mathbb{E}\left{g\left(X_t\right)\right}-g(x)}{t} \ & =\lim {t \downarrow 0} \mathbb{E}\left{\frac{1}{t} \int_0^t \mathcal{L} g\left(X_s\right) d s\right}=\mathcal{L} g(x) . \end{aligned}
The differential operator $\mathcal{L}$ given by (1.61) is called the infinitesimal generator of the Markov process $\left(X_t\right)$.

## 金融代写|波动率模型代写Market Volatility Modelling代考|Application to the Black-Scholes Partial Differential Equation

In the previous section we assumed the existence, uniqueness, and regularity of the solution of the partial differential equation (1.66) in order to apply Itô’s formula.. A sufficient condition for this is that the coefficients $\mu$ and $\sigma$ are regular enough and that the operator $\mathcal{L}_t$ is uniformly elliptic, meaning (in this one-dimensional situation) that there exists a positive constant $A$ such that
$$\sigma^2(t, x) \geq A>0 \quad \text { for every } t \geq 0 \text { and } x \in \mathcal{D},$$
so that the diffusion coefficient $\sigma^2(t, x)$ cannot become too small. Here $\mathcal{D}$ is the domain of the process $\left(X_t\right)$, which may be natural (e.g., $\mathcal{D}={x>0}$ for the geometric Brownian motion) or imposed externally from other modeling considerations.

When $\mu(t, x)=r x$ and $\sigma(t, x)=\sigma x$ in (1.66), we have the Black-Scholes partial differential equation (1.35) on the domain ${x>0}$. The ellipticity condition (1.68) is clearly not satisfied, since the diffusion coefficient $\sigma^2 x^2$ goes to zero as the state variable approaches zero. We get around this difficulty here (and also in more general situations) with the change of variable $P(t, x)=u(t, y=\log x)$, so that equation (1.35) becomes
$$\frac{\partial u}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 u}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial u}{\partial y}-r u=0$$
to be solved for $0 \leq t \leq T, y \in \mathbb{R}$, and with the final condition $u(T, y)=h\left(e^y\right)$. The operator
$$\mathcal{L}=\frac{1}{2} \sigma^2 \frac{\partial^2}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial}{\partial y}$$
is the infinitesimal generator of the (nonstandard) Brownian motion
$$Y_t=\left(r-\frac{1}{2} \sigma^2\right) t+\sigma W_t^{\star},$$
where $\left(W_t^{\star}\right)$ is a standard Brownian motion under $P^{\star}$. We use here the same notation as in the equivalent martingale measure context, but the only important fact is that $W^$ is a standard Brownian motion with respect to the probability used to compute the expectation in the Feynman-Kac formula (1.67). Applying this formula to $Y_t$ yields $$u(t, y)=\mathbb{E}^\left{e^{-r(T-t)} h\left(e^{y+\left(r-\sigma^2 / 2\right)(T-t)+\sigma\left(W_T^-W_t^\right)} \mid Y_t=y\right}\right.$$ which is indeed the same as (1.57) by undoing the change of variable $e^y=x$.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Infinitesimal Generators and Associated Martingales

$$d X_t=\mu\left(X_t\right) d t+\sigma\left(X_t\right) d W_t .$$

$$\mathcal{L} \mathcal{G}(x)-\frac{1}{2} \sigma^2(x) g^{\prime \prime}(x)+\mu(x) g^{\prime}(x)$$

$$d g\left(X_t\right)=\mathcal{L} g\left(X_t\right) d t+g^{\prime}\left(X_t\right) \sigma\left(X_t\right) d W_t$$

$$M_t=g\left(X_t\right)-\int_0^t \mathcal{L} g\left(X_s\right) d s$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Application to the Black-Scholes Partial Differential Equation

$$\sigma^2(t, x) \geq A>0 \quad \text { for every } t \geq 0 \text { and } x \in \mathcal{D},$$

$$\frac{\partial u}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 u}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial u}{\partial y}-r u=0$$

$$\mathcal{L}=\frac{1}{2} \sigma^2 \frac{\partial^2}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial}{\partial y}$$

$$Y_t=\left(r-\frac{1}{2} \sigma^2\right) t+\sigma W_t^{\star},$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|波动率模型代写Market Volatility Modelling代考|FM321

statistics-lab™ 为您的留学生涯保驾护航 在代写波动率模型Volatility Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写波动率模型Volatility Modelling代写方面经验极为丰富，各种代写波动率模型Volatility Modelling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Pricing

We mentioned in Section 1.2.1 that, unless $\mu=r$, the expected value under the $s u b$ jective probability $\mathbb{P}$ of the discounted payoff of a derivative (1.23) would lead to an opportunity for arbitrage. This is closely related to the fact that the discounted price $\widetilde{X}_t=e^{-r t} X_t$ is not a martingale since, from (1.18),
$$d \tilde{X}_t=(\mu-r) \tilde{X}_t d t+\sigma \tilde{X}_t d W_t,$$
which contains a nonzero drift term if $\mu \neq r$.
The main result we want to build in this section is that there is a unique probability measure $\mathbb{P}^$ equivalent to $\mathbb{P}$ such that, under this probability, (i) the discounted price $\widetilde{X}_t$ is a martingale and (ii) the expected value under $\mathbb{P}^$ of the discounted payoff of a derivative gives its no-arbitrage price. Such a probability measure describing a risk-neutral world is called an equivalent martingale measure.

In order to find a probability measure under which the discounted price $\tilde{X}t$ is a martingale, we rewrite (1.43) in such a way that the drift term is “absorbed” into the martingale term: We set $$d \widetilde{X}_t=\sigma \widetilde{X}_t\left[d W_t+\left(\frac{\mu-r}{\sigma}\right) d t\right] .$$ $$\theta=\frac{\mu-r}{\sigma},$$ called the market price of asset risk, and define $$W_t^{\star}=W_t+\int_0^t \theta d s=W_t+\theta t,$$ so that $$d \tilde{X}_t=\sigma \tilde{X}_t d W_t^{\star} .$$ Using the characterization (1.3), it is easy to check that the positive random variable $\xi_T^\theta$ defined by $$\xi_T^\theta=\exp \left(-\theta W_T-\frac{1}{2} \theta^2 T\right)$$ has an expected value (with respect to $\mathbb{P}$ ) equal to 1 (the Cameron-Martin formula). More generally, it has a conditional expectation with respect to $\mathcal{F}_t$ given by $$\mathbb{E}\left{\xi_T^\theta \mid \mathcal{F}_t\right}=\exp \left(-\theta W_t-\frac{1}{2} \theta^2 t\right)=\xi_t^\theta \quad \text { for } 0 \leq t \leq T,$$ which defines a martingale denoted by $\left(\xi_t^\theta\right){0 \leq t \leq T}$.
We now introduce the probability measure $\mathbb{P}^{\star}$. It is an equivalent measure to $\mathbb{P}$, meaning that it has the same null sets $\left(\mathbb{P}^*\right.$ and $\mathbb{P}$ agree on which events have zero probability); $\mathbb{P}^{\star}$ has the density $\xi_T^\theta$ with respect to $\mathbb{P}$,
$$d \mathbb{P}^{\star}=\xi_T^\theta d \mathbb{P} .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Expectations and Partial Differential Equations

In Section 1.4.4 we used the Markov property of the stock price $\left(X_t\right)$ and, in order to compute $X_T$ knowing that $X_t=x$ at time $t \leq T$, we solved the stochastic differential equation (1.2) between $t$ and $T$. This was a particular case of the general situation where $\left(X_t\right)$ is the unique solution of the stochastic differential equation (1.11). We denote by $\left(X_s^{t, x}\right){s \geq t}$ the solution of that equation, starting from $x$ at time $t$ : $$X_s^{t, x}=x+\int_t^s \mu\left(u, X_u^{t, x}\right) d u+\int_t^s \sigma\left(u, X_u^{t, x}\right) d W_u,$$ and we assume enough regularity in the coefficients $\mu$ and $\sigma$ for $\left(X_s^{t, x}\right)$ to be jointly continuous in the three variables $(t, x, s)$. The flow property for deterministic differential equations can be extended to stochastic differential equations like (1.11); it says that, in order to compute the solution at time $s>t$ starting at time 0 from point $x$, one can use $$x \longrightarrow X_t^{0, x} \longrightarrow X_s^{1, X_t^{0, x}}=X_s^{0, x} \quad(\mathbb{P} \text {-a.s.). }$$ In other words, one can solve the equation from 0 to $t$, starting from $x$, to obtain $X_t^{0, x}$. Then we solve the equation from $t$ to $s$, starting from $X_t^{0, x}$. This is the same as solving the equation from 0 to $s$, starting from $x$. The Markov property is a consequence and can be stated as follows: $$\mathbb{E}\left{h\left(X_s\right) \mid \mathcal{F}_1\right}=\left.\mathbb{E}\left{h\left(X_s^{t \cdot x}\right)\right}\right|{x=X_1},$$
which is what we have used with $s=T$ to derive (1.55). Observe that the discounting factor could be pulled out of the conditional expection since the interest rate is constant (not random). In the time-homogeneous case ( $\mu$ and $\sigma$ independent of time) we further have $$\mathbb{E}\left{h\left(X_s^{t, X}\right)\right}=\mathbb{E}\left{h\left(X_{s-t}^{0, X}\right)\right},$$
which could have been used with $s=T$ to derive (1.57) since $W_{T-t}^*$ is $\mathcal{N}(0, T-t)$ distributed.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Pricing

$$d \tilde{X}_t=(\mu-r) \tilde{X}_t d t+\sigma \tilde{X}_t d W_t,$$

$$\begin{gathered} d \widetilde{X}_t=\sigma \widetilde{X}_t\left[d W_t+\left(\frac{\mu-r}{\sigma}\right) d t\right] . \ \theta=\frac{\mu-r}{\sigma}, \end{gathered}$$

$$W_t^{\star}=W_t+\int_0^t \theta d s=W_t+\theta t$$

$$d \tilde{X}_t=\sigma \tilde{X}_t d W_t^{\star} .$$

$$\xi_T^\theta=\exp \left(-\theta W_T-\frac{1}{2} \theta^2 T\right)$$

$$d \mathbb{P}^{\star}=\xi_T^\theta d \mathbb{P} .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Expectations and Partial Differential Equations

$$X_s^{t, x}=x+\int_t^s \mu\left(u, X_u^{t, x}\right) d u+\int_t^s \sigma\left(u, X_u^{t, x}\right) d W_u,$$

$$x \longrightarrow X_t^{0, x} \longrightarrow X_s^{1, X_t^{0, x}}=X_s^{0, x} \quad(\mathbb{P} \text {-a.s. }) .$$

Imathbb ${E} \backslash$ eft $\left{h \backslash l\right.$ eft $\left(X_{-} s^{\wedge}{t, X} \backslash\right.$ ight $\left.) \backslash r i g h t\right}=\backslash m a t h b b{E} \backslash l$ eft $\left{h \backslash l\right.$ eft $\left(X_{-}{\right.$st $} \wedge{0, X} \backslash$ ight $) \backslash$ 正确的 $}$ ，

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|波动率模型代写Market Volatility Modelling代考|FE720

statistics-lab™ 为您的留学生涯保驾护航 在代写波动率模型Volatility Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写波动率模型Volatility Modelling代写方面经验极为丰富，各种代写波动率模型Volatility Modelling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Replicating Strategies

The Black-Scholes analysis of a European-style derivative yields an explicit trading strategy in the underlying risky asset and riskless bond whose terminal payoff is equal to the payoff $h\left(X_T\right)$ of the derivative at maturity, no matter what path the stock price takes. Thus, selling the derivative and holding a dynamically adjusted portfolio according to this strategy “covers” an investor against all risk of eventual loss, because a loss incurred at the final time from one part of this portfolio will be exactly compensated by a gain in the other part. This replicating strat$e g y$, as it is known, therefore provides an insurance policy against the risk of being short the derivative. It is called a dynamic hedging strategy since it involves continuous trading, where to hedge means to eliminate risk. The essential step in the Black-Scholes methodology is the construction of this replicating strategy and arguing, based on no arbitrage, that the value of the replicating portfolio at time $t$ is the fair price of the derivative. We develop this argument in the following sections.

We consider a European-style derivative with payoff $h\left(X_T\right)$, a function of the underlying asset price at maturity time $T$. Assume that the stock price $\left(X_t\right)$ follows the geometric Brownian motion model (1.20), a solution of the stochastic differential equation (1.2). A trading strategy is a pair $\left(a_t, b_t\right)$ of adapted processes specifying the number of units held at time $t$ of the underlying asset and the riskless bond, respectively. We suppose that $\mathbb{E}\left{\int_0^T\left(a_t\right)^2 d t\right}$ and $\int_0^T\left|b_t\right| d t$ are finite so that the stochastic integral involving $\left(a_t\right)$ and the usual integral involving $\left(b_t\right)$ are well-defined.

Assuming, as in (1.1), that the price of the bond at time $t$ is $\beta_t=e^{r t}$, the value at time $t$ of this portfolio is $a_t X_t+b_t e^{\prime t}$. It will replicate the derivative at maturity if its value at time $T$ is almost surely equal to the payoff:
$$a_T X_T+b_T e^{r T}=h\left(X_T\right) .$$
In addition, this portfolio is to be self-financing, meaning that the variations of its value are due only to the variations of the market – that is, the variations of the stock and bond prices. No further funds are required after the initial investment,yields an instant profit with no exposure to future loss, since the terminal payoff of the trading strategy is equal to the payoff of the derivative.

## 金融代写|波动率模型代写Market Volatility Modelling代考|Self-Financing Portfolios

As in Section 1.3.1, a portfolio comprises $a_i$ units of stock and $b_t$ in bonds; we denote by $V_t$ its value at time $t$ :
$$V_t=a_t X_t+b_t e^{r t} .$$
The self-financing property (1.28), namely $d V_t=a_t d X_t+r b_t e^{r t} d t$, implies that the discounted value of the portfolio, $\widetilde{V}_t=e^{-r t} V_t$, is a martingale under the risk-neutral probability $\mathbb{P}^{\star}$. This important property of self-financing portfolios is cbtained as follows:
\begin{aligned} d \tilde{V}_t & =-r e^{-r t} V_t d t+e^{-r t} d V_t \ & =-r e^{-r t}\left(a_t X_t+b_t e^{r t}\right) d t-e^{-r t}\left(a_t d X_t+r b_t e^{r t} d t\right) \ & =-r e^{-r t} a_t X_t d t+e^{-r t} a_t d X_t \ & =a_t d\left(e^{-r t} X_t\right) \ & =a_t d \tilde{X}_t \ & =\sigma a_t \widetilde{X}_t d W_t^{\star} \quad(\text { by }(1.46)), \end{aligned}
which shows that $\left(\tilde{V}_t\right)$ is a martingale under $\mathbb{P}^{\star}$ as a stochastic integral with respect to the Brownian motion $\left(W_t^*\right)$. Indeed, the same computation shows that if a portfolio satisfies $d \tilde{V}_t=a_t d \widetilde{X}_t$ then it is self-financing.

A simple calculation demonstrates the connection between martingales and no arbitrage. Suppose that $\left(a_t, b_t\right)_{0 \leq t \leq T}$ is a self-financing arbitrage strategy; that is,
$$V_T \geq e^{r T} V_0 \quad(I P \text {-a.s. }),$$
with
$$\mathbb{P}\left{V_T>e^{r T} V_0\right}>0,$$
so that the strategy never makes less than money in the bank and there is some chance of making more. But
$$\mathbb{E}^\left{V_T\right}=e^{r T} V_0$$ by the martingale property, so (1.51) and (1.52) cannot hold. This is because $\mathbb{P}$ and $\mathbb{P}^$ are equivalent and so (1.51) and (1.52) also hold with $\mathbb{P}$ replaced by $\mathbb{P}^{\star}$.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Replicating Strategies

$$a_T X_T+b_T e^{r T}=h\left(X_T\right) .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Self-Financing Portfolios

$$V_t=a_t X_t+b_t e^{r t} .$$

$$V_T \geq e^{r T} V_0 \quad(I P \text {-a.s. }),$$

Imathbb ${P} \backslash l$ eft $\left{V_{-} T>e^{\wedge}{r \mathrm{~T}} V_{-} \backslash \backslash\right.$ ight $}>0$,

Imathbb{E $}^{\wedge} \backslash l$ eft $\left{V_{-} T \backslash r i g h t\right}=e \wedge{r ~ T} \vee_{-} 0$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。